Space Elevators & Tethers
The biggest issue would be getting materials on/off the planet. Currently it costs $20-25k per pound to put something into orbit. It's possible, but far from affordable...
I just happen to listen to a podcast from .Net Rocks talking about Space Elevators and Tethers released earlier this month.
We could have Space Elevators by 2035 and a Japanese Company is aiming to build one by 2050...
Even assuming it takes longer, I think it's easy to say that we could be easily moving stuff up and down space elevators by 2100.
Once we can do that, the cost of moving stuff up and out should drop from 20,000.00 to 200.00 per pound - and then lower as time goes on. It then becomes insanely affordable (compared to now) to move into orbit and back.
Your planets aren't that far apart (per comment, they are closer than us and the moon.), it's not that far of a stretch to take some of our puny space craft - add a few decades of research and improvements - and bam... easy to see trade blooming between the two planets: Just like it's easy to see a Moon colony being realistic and a Mars Colony not far behind once it costs 1/100th of what it costs now.
EDIT: Using a Lagrange Point
As per comments, I think it would be realistic - assuming we have progressed to the point to create carbon nano tubes to make a "Space Elevator" realistic - to assume that in 100 years or less you could create an elevator between the two worlds.
Between the two worlds should exist some sort of Lagrange Point - a balancing point of gravity. Place a station between the two worlds at that point. Then extend filaments towards each world. At some point, you have a platform at the end of each filament. One on each world - with a station in between. Assuming the rotations and equators don't match perfectly, you still have a suspended platform on each world. (If the equators/rotations match... then you just anchor them to the ground on each world... but I don't think the odds favor that scenario)
Now you don't even have to use rockets to get between the worlds... Elevator go up... Elevator go down... your standing on the other planet.